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In this paper we prove that $E$ admits a unique $\\omega$-balanced metric if and only if $\\frac{r_j}{N_j}=\\frac{r_k}{N_k}$ for all $j, k=1, ..., m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\\dim H^0(M, E_j)$. We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety $(M, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.3078","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-01-16T16:38:14Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"edd925071ef752b7ec09f96bbc76615d633439ec134eb97f2756f9c305a7726a","abstract_canon_sha256":"0cfd3295033a64e5820269683bd0195febcea57842ea2b44a1d283e325a12e35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:39.501146Z","signature_b64":"UZNEY4DD6fazC/B7Bzwy0p5wxiLDikzA9mduVAjpmaVqIv5NrfxNVm5YLS7W0w+VH4ugitQ4A20bN0ZJAyCmDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff88147b60593641940d2e43f8fb3468e03d892ca82caf8823a8c1e6801a66b8","last_reissued_at":"2026-05-18T02:03:39.500733Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:39.500733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Balanced metrics on homogeneous vector bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AG","authors_text":"Roberto Mossa","submitted_at":"2011-01-16T16:38:14Z","abstract_excerpt":"Let $E\\rightarrow M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, \\omega)$ and let $E=E_1\\oplus... \\oplus E_m\\rightarrow M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $\\omega$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique $\\omega$-balanced metric if and only if $\\frac{r_j}{N_j}=\\frac{r_k}{N_k}$ for all $j, k=1, ..., m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\\dim H^0(M, E_j)$. 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